Some approaches to a conjecture on short cycles in digraphs

We consider the following special case of a conjecture due to Caccetta and Haggkvist: Let be a digraph on vertices that all have in-degree and out-degree at least . Then contains a directed cycle of length 2 or 3. We discuss several necessary conditions for possible counterexamples to this conjecture, in terms of cycle structure, diameter, maximum degree, clique number, toughness, and local structure. These conditions have not enabled us to prove or refute the conjecture, but they lead to proofs of special instances of the conjecture.